Theorem toponcom 22968

Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
toponcom	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

Proof of Theorem toponcom

Step	Hyp	Ref	Expression
1		toponuni 22954	. . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐽) → ∪ 𝐽 = ∪ 𝐾)
2	1	eqcomd 2767	. . 3 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐽) → ∪ 𝐾 = ∪ 𝐽)
3	2	anim2i 626	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → (𝐽 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐽))
4		istopon 22952	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ (𝐽 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐽))
5	3, 4	sylibr 236	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∪ cuni 4864  ‘cfv 6517  Topctop 22933  TopOnctopon 22950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-topon 22951

This theorem is referenced by:  toponcomb  22969  kgencn3  23598